Worksheet: Equation of a Plane: Vector, Scalar, and General Forms

In this worksheet, we will practice finding the vector, scalar (standard or component), and general (Cartesian or normal) forms of the equation of a plane given the normal vector and a point on it.

Q1:

Write, in normal form, the equation of the plane (1,0,3), (1,2,−1) and (6,1,6).

A𝑥+3𝑧+20=0

B𝑥+3𝑧−20=0

C𝑥−2𝑦−𝑧−4=0

D𝑥−2𝑦−𝑧−2=0

E𝑥−2𝑦−𝑧+2=0

Q2:

Which of the following planes contains the straight line rijkijk=−6−−4+𝑡(4+−)?

Arijk⋅(2−6+15)=−31

Brijk⋅(2−6+15)=0

Crijk⋅(2−3+5)=−29

Drijk⋅(2−3+5)=58

Erijk⋅(4+−)=0

Q3:

Find the vector form of the equation of the plane that has normal vector nijk=++
and contains the point (2,6,6).

A⟨1,1,1⟩⋅=14r

Br=14

C⟨1,1,1⟩⋅=⟨2,6,6⟩r

Dr=⟨2,6,6⟩

Q4:

Find the direction cosines of the normal to the plane 4𝑥+8𝑦−3𝑧=28.

A4√8989,8√8989,−3√8989

B489,889,−389

C4√1515,8√1515,−√155

D14,12,−316

Q5:

In which of the following planes does the point (3,−1,5) lie?

A2𝑥+𝑦−2𝑧+23=0

B𝑥−2𝑦+2𝑧−15=0

C2𝑥−4𝑦+𝑧+5=0

D−4𝑥−4𝑦+2𝑧+7=0

E3𝑥−𝑦+5𝑧=0

Q6:

Which of the following points lies in the plane 3(𝑥+4)−2(𝑦+1)−7(𝑧−6)=0?

A(−4,−1,6)

B(3,−2,−7)

C(7,−1,−13)

D(4,1,−6)

Q7:

Find the general equation of the plane which passes through the point (3,−8,−7) and contains the 𝑥-axis.

A3𝑥−7𝑦+8𝑧=0

B3𝑥−8𝑦−7𝑧=0

C−7𝑥+8𝑧=0

D8𝑥−7𝑦=0

E−7𝑦+8𝑧=0

Q8:

Find the equation of the plane 𝑥𝑦.

A𝑥+𝑦=0

B𝑥+𝑦=𝑧

C𝑧=0

D𝑧−𝑥𝑦=0

E𝑥=𝑦

Q9:

Find the equation of the plane which is perpendicular to the vector
Aijk=5−7−3
and passes through the point 𝐵(−5,5,9).

A−5𝑥+5𝑦+9𝑧−5=0

B5𝑥−7𝑦−3𝑧+87=0

C−5𝑥+5𝑦+9𝑧+87=0

D5𝑥−7𝑦−3𝑧−5=0

E5𝑥−7𝑦−3𝑧−87=0

Q10:

A plane passes through (−2,−2,3) and has normal ⟨−4,1,−4⟩. Give its equation in vector form.

Ar=−6

Br=⟨−4,1,−4⟩

C⟨−4,1,−4⟩⋅=−6r

D⟨−4,1,−4⟩⋅=⟨−2,−2,3⟩r

Q11:

Which of the following does the equation −7𝑥−2𝑧=0 represent in three-dimensional space?

Aa plane containing the 𝑦-axis

Ba plane containing the 𝑧-axis

Ca straight line whose direction ratios are (−7,0,−2)

Da plane containing the 𝑥-axis

Q12:

Determine the general form of the equation for a plane in which the two straight lines 𝐿∶𝑥+8−7=𝑦+7−5=𝑧+53 and 𝐿∶𝑥+84=𝑦+73=𝑧+54 lie.

A−29𝑥−40𝑦+𝑧−43=0

B4𝑥+3𝑦+4𝑧+146=0

C−29𝑥+40𝑦−𝑧+43=0

D−7𝑥−5𝑦+3𝑧−76=0

Q13:

Determine the Cartesian equation of the straight line passing through the point (−2,9,2) that is perpendicular to the plane 5𝑥−6𝑦−6𝑧−11=0.

A𝑥+5−2=𝑦−69=𝑧−62

B𝑥−25=𝑦+9−6=𝑧+2−6

C𝑥−5−2=𝑦+69=𝑧+62

D𝑥+25=𝑦−9−6=𝑧−2−6

Q14:

To which of the following planes is the straight line 𝑥−24=𝑦+7−3=𝑧+96 perpendicular?

A12𝑥−9𝑦+18𝑧−19=0

B4𝑥−14𝑦−18𝑧+19=0

C2𝑥−7𝑦−9𝑧=0

D4𝑥+3𝑦+6𝑧=−19

Q15:

Find the general equation of the plane which passes through the two
points 𝐴(8,−7,−2) and 𝐵(1,−4,−1),
given that the distance from the 𝑥-intercept to the origin is equal to the distance from the 𝑦-intercept to the origin.

A−7𝑥−7𝑦−74𝑧−1=0

B4𝑥+4𝑦+𝑧+7=0

C−74𝑥−74𝑦−7𝑧+1=0

D𝑥+𝑦+4𝑧+7=0

Q16:

Write, in normal form, the equation of the plane P containing the point
Q=⟨5,1,−2⟩ and perpendicular to the vector n=⟨4,−4,3⟩.

A4𝑥−4𝑦+3𝑧−10=0

B4𝑥−4𝑦+3𝑧+10=0

C4𝑥−4𝑦+3𝑧+4=0

D5𝑥+𝑦−2𝑧+10=0

E5𝑥+𝑦−2𝑧−10=0

Q17:

Find the equation, in vector form, of the plane passing through the points
(1,2,2),
(3,1,−4), and
(0,3,3).

Ar=⟨5,4,1⟩

Br=15

C(5,4,1)⋅=⟨1,2,2⟩r

D⟨5,4,1⟩⋅=15r

Q18:

Find the vector form of the equation of the plane containing the two straight lines
rijkijk=(−−3)+𝑡(3+3+4) and rijkijk=(−−2−3)+𝑡(−−2−4).

A⟨4,−8,3⟩⋅=−3r

B⟨4,−4,3⟩⋅=−1r

C⟨4,−8,3⟩⋅=3r

D⟨20,16,9⟩⋅=−23r

Q19:

Which of the following is the equation of a plane that bisects the line segment between the two points (4,−2,−6) and (8,4,2)?

A𝑥+𝑦−𝑧+5=0

B𝑥−𝑦−𝑧−5=0

C𝑥−𝑦+𝑧+5=0

D𝑥+𝑦+𝑧−5=0

Q20:

Given that ⃖⃗𝐴𝐵 is parallel to the plane 8𝑥−5𝑦−2𝑧−5=0,
where the coordinates of 𝐴 and 𝐵 are (−4,3,𝑚) and (−3,−3,𝑛), respectively, find the value of (𝑛−𝑚).

Q21:

Find the Cartesian equation of the plane (𝑥,𝑦,𝑧)=(−7,−5,−3)+𝑡(−3,−8,1)+𝑡(2,1,3), where
𝑡 and 𝑡 are parameters.

A25𝑥−11𝑦−13𝑧+81=0

B−3𝑥−8𝑦+𝑧−58=0

C−7𝑥−5𝑦−3𝑧+11=0

D2𝑥+𝑦+3𝑧+28=0

E𝑥−7𝑦−4𝑧+30=0

Q22:

Write, in normal form, the equation of the plane containing (−3,1,−3), (4,−4,3), and (0,0,1).

A−14𝑥−10𝑦+8𝑧+56=0

B−14𝑥−10𝑦+8𝑧+8=0

C−3𝑥+𝑦−3𝑧−8=0

D−14𝑥−10𝑦+8𝑧−8=0

E−3𝑥+𝑦−3𝑧+8=0

Q23:

Find the general equation of the plane which contains the straight line 𝑥−1−7=𝑦+84=𝑧+34 and the point 𝐴(−8,−4,3).

A−7𝑥+4𝑦+4𝑧−52=0

B4𝑥−3𝑦+4𝑧−16=0

C𝑥−8𝑦−3𝑧−15=0

D−7𝑥+4𝑦+4𝑧+51=0

E4𝑥+3𝑦+4𝑧+32=0

Q24:

Find the general equation of the plane which passes through the two points (−6,9,8) and (−8,5,−2) and is parallel to the vector A=⟨2,1,3⟩.

A−6𝑥+9𝑦+8𝑧−77=0

B2𝑥+𝑦+3𝑧−21=0

C−𝑥−7𝑦+3𝑧+33=0

D−𝑥+7𝑦+3𝑧−93=0

E2𝑥+𝑦+3𝑧+17=0

Q25:

Find the general equation of the plane passing through the point 𝐴(7,5,−3) and perpendicular to the straight line passing through the two points 𝐵(−5,−7,−1) and 𝐶(9,−9,7).